I'm having trouble with the following: I have made the following development graph ('total') based on some data from my job.
I would like to find a function that approximates this graph. The idea what I wanted to do was to find some general function with some parameters that I can vary to closely approximate the function.
I have tried to use to following function $$a(x-b)e^{-cx}+1$$ where I can vary $a,b$ and $c$. However, if I use least squares to find good values of $a,b$ and $c$, the best I can get is what is shown as 'Theoretical' in the image, so it seems like this function can't really approximate the shape very well. In excel, I made a polynomial trendline that looks pretty much like what I want, but ofcourse it doesn't have the desired long term behaviour.
Can anybody help me to modify my function so that varying the parameters would yield a better result (preferably with as few parameters as possible).
Given that it is a development pattern, the only property it really has is that should tend to 1 as $x\rightarrow \infty$. I guess it should also start at $0$ for $x=0$, however that is not really a requirement for what I'm going to use it for, so that is why my intitial function doesn't necessarily have that property.
You should be able to have a result similar to your dotted line with your model. Maybe the solver you are using is not made to work with exponential functions. By manualy playing with your model, I managed to get something close to what you are looking for (https://www.desmos.com/calculator/cprgnujmsx)
But in order to have the best values, you need to use a solver that handles this type of functions.
Another possibility, is to use a different model like this one:
$$f(x) = A\exp(-ax)-B\exp(-bx)+C$$ with $$A \gt 0, B \gt 0,a\gt 0, b\gt a$$
You can reduce the number of parameters by setting constraints such as the asymptotic value (given by the constant $C$). So, in your case, $C=1$. If you need the function to pass by the origin, for example, you would set $B=A+C$
You can play with this graph to get an idea of what the constants do on the function : https://www.desmos.com/calculator/yu4vw2eh7f
The advantages of this model, is that you only have exponentials.